As a thought experiment let us assume that we have isolated a magnetic domain. This domain is of finite size and we know its dimensions. Assuming that we can measure an infinitesimal field, will there be a certain region beyond which the field won't be applicable?

The instinctive answer to this question is no, but if you think about it we see the magnet's influence on the space around it as the result of equipotential regions then the contention is that only so many discrete equipotential regions are possible (the fact that something is not countable doesn't automatically mean it's infinite). Hence, that line of thought goes, there should be a limit theoretically and practically until which a field can exist.

Can you please clarify this sticking point for me? Am I pushing the analogies we use to understand fields too far? What conceptual mistake am I making over here?

3 Answers
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What you are suggesting is that there is a quantization law for magnetic fields, so that if the field is too small, it is actually exactly zero. This is not true in quantum electrodynamics, the field shrinks to zero as a power law, and the influence of the magnet is felt arbitrarily far out.

But the experiment which you need to measure the field becomes larger and larger. Given a particle of charge e, you can take it in a loop around the region with a magnetic field, to see interference fringes which depend on the area of the loop according to the phase law

$$\delta \phi = \oint A dl = \int B dA $$

To get an equal phase change, you need to make a bigger loop when you are far enough out.

The question of the measurability of tiny fields is not purely academic. The phase method of detecting fields is extremely sensitive to tiny fields when you use a superconducting loop and measure magnetic fields by the phase the current gets around the loop, this is a SQUID. The SQUID can measure fields which would be too small to measure other ways, and it can be adjusted to measure fields that are infinitesimal with regards to less sensitive detectors.

The philosophical position that all real quantities must be eventually discrete is not particularly useful, because the real quantities can arise from large-system-limits, and so be arbitrarily fine-grained as you get to larger and larger system sizes. For example, if the temperature is arbitrarily small, is it indistinguishable from exactly zero temperature? This depends on the size of the system. If you make the system bigger, you can see the difference from zero temperature even finer.

The limit of large number of photon exchanges, in a Feynman particle view of the field, is the large number limit that makes a tiny field make sense. If the field is too tiny, you will only get a finite number of photons affecting your device, and the effects vanish. But if you make the device bigger, you become sensitive to more photons. These type of large N quantities can philosophically be real valued without any contradiction, because the grain-size for the real-number quantity is physical, and determined by the parameters of the system and measuring device.

we share a name and an interest in physics, so I thought I might give you my POV.

a) There are classical fields for which I agree with the answer of Edgar Bonet, that the crux is in "infinitesimal".

b) When we reach what we call infinitesimals for physical quantities we enter the realm of quantum mechanics.

Even though physics is not a consensus science, still there is a dominant school based on all experimental results up to now. This school, to which most physicists adhere including myself, says that nature is basically quantum mechanical and macroscopic field equations are the convolution of the real quantum mechanical functions that dominate microscopically. These equations do not contain classical fields, but the carriers of what become fields macroscopically. In the case of the magnetic field the carrier is the photon, and any interaction, which includes measurements, will happen through photon exchanges .

If you are at infinity, i.e. very far away from the macroscopic magnetic domain you want to examine in your thought experiment, the photons that will tell you that the domain exists will take years/aeons to reach the detector and back to you ( special relativity, think of stars) and the probabilities for this to happen will be very low but still, sometime in infinite time, still non zero.

One can intuit how the convolutions ( successive integrations over more inclusive variables) can generate the macroscopic electric and magnetic fields from the virtual exchanges of photons: photons can be both particles and waves. As waves their fields can be alined because they can be polarized and act in uniform to generate a specific field, but it is hand waving I am doing here.
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anna vDec 22 '11 at 4:38